Differentially Private Stochastic Gradient Descent (DP-SGD) is the dominant paradigm for private training, but its fundamental limitations under worst-case adversarial privacy definitions remain poorly understood. We analyze DP-SGD in the $f$-differential privacy framework, which characterizes privacy via hypothesis-testing trade-off curves, and study shuffled sampling over a single epoch with $M$ gradient updates. We derive an explicit suboptimal upper bound on the achievable trade-off curve. This result induces a geometric lower bound on the separation $κ$ which is the maximum distance between the mechanism's trade-off curve and the ideal random-guessing line. Because a large separation implies significant adversarial advantage, meaningful privacy requires small $κ$. However, we prove that enforcing a small separation imposes a strict lower bound on the Gaussian noise multiplier $σ$, which directly limits the achievable utility. In particular, under the standard worst-case adversarial model, shuffled DP-SGD must satisfy $σ\ge \frac{1}{\sqrt{2\ln M}}$ $\quad\text{or}\quad$ $κ\ge\ \frac{1}{\sqrt{8}}\!\left(1-\frac{1}{\sqrt{4π\ln M}}\right)$, and thus cannot simultaneously achieve strong privacy and high utility. Although this bound vanishes asymptotically as $M \to \infty$, the convergence is extremely slow: even for practically relevant numbers of updates the required noise magnitude remains substantial. We further show that the same limitation extends to Poisson subsampling up to constant factors. Our experiments confirm that the noise levels implied by this bound leads to significant accuracy degradation at realistic training settings, thus showing a critical bottleneck in DP-SGD under standard worst-case adversarial assumptions.

翻译：差分私有随机梯度下降（DP-SGD）是私有训练的主流范式，但其在最坏情况对抗性隐私定义下的基本限制仍不明确。我们在$f$-差分隐私框架下分析DP-SGD，该框架通过假设检验权衡曲线刻画隐私特性，并研究了在单周期内进行$M次梯度更新的混洗采样策略。我们推导出可达权衡曲线的显式次优上界，该结果引出了分离度κ的几何下界——即机制权衡曲线与理想随机猜测线之间的最大距离。由于大分离度意味着显著对抗优势，有意义的隐私保护需要较小的κ值。然而，我们证明强制实现小分离度会对高斯噪声乘子σ施加严格下界，这将直接限制可达效用。具体而言，在最坏情况对抗模型下，混洗DP-SGD必须满足σ≥1/√(2ln M)或κ≥1/√8·(1-1/√(4π ln M))，因此无法同时实现强隐私保护与高可用性。虽然当M→∞时该界限渐近消失，但收敛速度极慢：即使在实践中常见的更新次数下，所需噪声幅度仍然显著。我们进一步证明该限制可扩展至泊松子采样（仅相差常数因子）。实验证实，该界限隐含的噪声水平在现实训练设置下会导致显著的准确率下降，从而揭示了标准最坏情况对抗假设下DP-SGD的关键瓶颈。